11th Class Mathematics Conic Sections Question Bank Critical Thinking

  • question_answer The line \[x\cos \alpha +y\sin \alpha =p\] will be a tangent to the conic \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\], if        [Roorkee 1978]

    A)            \[{{p}^{2}}={{a}^{2}}{{\sin }^{2}}\alpha +{{b}^{2}}{{\cos }^{2}}\alpha \]

    B)            \[{{p}^{2}}={{a}^{2}}+{{b}^{2}}\]

    C)            \[{{p}^{2}}={{b}^{2}}{{\sin }^{2}}\alpha +{{a}^{2}}{{\cos }^{2}}\alpha \]

    D)            None of these

    Correct Answer: C

    Solution :

               \[y=-x\cot \alpha +\frac{p}{\sin \alpha }\]is tangent to \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1,\] if \[\frac{p}{\sin \alpha }=\pm \sqrt{{{b}^{2}}+{{a}^{2}}{{\cot }^{2}}\alpha }\] or \[{{p}^{2}}={{b}^{2}}{{\sin }^{2}}\alpha +{{a}^{2}}{{\cos }^{2}}\alpha \].

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